Question

challenge the vertices of δabc are a(-5,5), b(-2,4), and c(-4,2). δabc is reflected across the y - axis and then reflected across the x - axis to produce the image δabc. graph δabc and δabc. which graph shows δabc and δabc? a. image of graph a b. image of graph b c. image of graph c

Explanation:

Step1: Reflect over y - axis

The rule for reflecting a point \((x,y)\) over the \(y\) - axis is \((x,y)\to(-x,y)\).
For point \(A(-5,5)\), after reflection over \(y\) - axis: \(A_1=(5,5)\)
For point \(B(-2,4)\), after reflection over \(y\) - axis: \(B_1=(2,4)\)
For point \(C(-4,2)\), after reflection over \(y\) - axis: \(C_1=(4,2)\)

Step2: Reflect over x - axis

The rule for reflecting a point \((x,y)\) over the \(x\) - axis is \((x,y)\to(x, - y)\).
For point \(A_1(5,5)\), after reflection over \(x\) - axis: \(A'=(5,-5)\)
For point \(B_1(2,4)\), after reflection over \(x\) - axis: \(B'=(2,-4)\)
For point \(C_1(4,2)\), after reflection over \(x\) - axis: \(C'=(4,-2)\)

Now we analyze the options:

• Option A: The original triangle \(\triangle ABC\) has vertices in the second quadrant (negative \(x\), positive \(y\)). After reflection over \(y\) - axis (to first quadrant) and then \(x\) - axis (to fourth quadrant). The coordinates of the final image \(\triangle A'B'C'\) should have positive \(x\) and negative \(y\) values. Let's check the positions. The original \(\triangle ABC\) with \(A(-5,5)\), \(B(-2,4)\), \(C(-4,2)\) is in the second quadrant. After \(y\) - axis reflection, it goes to first quadrant (\(A_1(5,5)\), \(B_1(2,4)\), \(C_1(4,2)\)) and then \(x\) - axis reflection to fourth quadrant (\(A'(5,-5)\), \(B'(2,-4)\), \(C'(4,-2)\)). Option A seems to have the original in second and image in fourth.
• Option B: The image after reflections does not match the calculated coordinates. The final image points should have positive \(x\) and negative \(y\), but the red triangle in B is not in the correct fourth - quadrant position with respect to the original.
• Option C: The positions of the triangles do not match the reflection rules. The original is in second, but the image is not in the correct fourth - quadrant position.

Answer:

A